The Speed Math Bible.pdf

The Speed Math Bible

written by Yamada Takumi, with the special collaboration of Danilo Lapegna

Transform your brain into an electronic calculator and master the mathematical strategies to triumph in every challenge!

"The 101 bibles" series

I - The Speed Math Bible

I'm quite sure we could all agree about a fact: the traditional way to teach mathematics has a lot of structural problems, and in most of cases it doesn't really help students to get confident with the subject. In fact, too many students finish their schooling still having a real "mathematical illiteracy", united with a burning hatred of everything concerning numbers and operations.

In particular, talking about classic method of teaching mathematics, I'm strongly of the opinion that:

It's really poor incentive for individual creativity: in fact, too many times

school will teach you that the proper method for performing a set of calculations i s "rigidly" one, and that everything should always be made in the same way. This obviously can't do much more than boring every student and generating the feeling that the matter itself, rather than improving one's mental skills, actually shrinks them, gradually transforming him/her into something that's more similar to an industrial machine.

This book, however, is designed to go far beyond this restricted vision and will teach you that the classic approach is not the only possible approach and that every set of mathematical calculations can be transformed into a deeply creative challenge.

The idea of mathematical "trick" is unjustly "demonised": very few people know that they could perform very complex calculations just by using extraordinarily rapid and effective numerical tricks.

And although it's widely accepted that learning the right balance between tricks and rigour, creativity and structure, quick solutions and harder solutions would be ideal for any student, schools usually continue to prefer following a "complexity at all costs" that of course does not nothing but alienating people from the matter.

It's promoted very little self-expression: too many people feel that they have

"nothing to share" with some mathematical concepts even because the rigidity of the taught method prevents everyone from expressing himself according to his talent, his wishes and his natural predispositions.

But this book presents a completely different approach, since everything among its pages will be deeply self-expression oriented, and the methods shown here will teach you not just one, unquestionable method to solve everything, but will give you the freedom to act according to what you feel to be easier and more compatible with your natural attitude.

It's promoted very little curiosity and "researching spirit": math, as taught

from some very bad teachers, is presented as a grey and squared world, made of endless repetitions and very few interesting things. But thanks to this book you will start an original journey through the singularities and the curiosities of the numbers world and you'll soon find out how many interesting and beautiful things are hidden into its deep harmony.

They speak very little about the real utility of mathematics: beyond the

ability to calculate your restaurant bill in an instant, the enormous advantage you will gain in every official test or exam, or the awesome progression of your logical and deductive faculties, you will discover that mathematics has extraordinary strategic and creative applications that will help you very often in your everyday challenges, giving you that extra oomph in the administration of your personal finance, your work, your studies, your health, your self-confidence and your strategic intelligence when dealing with any kind of challenge.

Math is also vital engine and fuel for every little technological wonder in the modern world. Computers and tablets could never exist today if two mathematicians like John von Neumann and Alan Turing didn't write the mathematical principles behind the first calculators. Internet would never have existed if no one had developed the mathematical principles the networks theory is based on. And the search engines and social networks could never be born without the equations and algorithms allowing each user to track, retrieve and organize documents, web pages and profiles directly from a set of chaotic data.

Moreover, the inventors of Google and revolutionaries of the modern world, Larry Page and Sergey Brin, are both graduates in mathematics.

What I'm trying to tell with this? Well, I'm not sure that every reader of this book will end up founding the new Google, but I'm sure that a greater mathematics competence in a world tuned on these frequencies can help each of us to be more a protagonist and less a bare spectator of it.

In other words, this book will not simply be a set of strategies for impressing someone or increasing your academic performance, but will let you start a journey through a pleasant and intriguing path of personal growth, along which you'll learn to be more effective, creative, confident and, why not, more intelligent.

In addition to that, I would like to give you a last advice: do not immediately try to learn every method explained here, but go slowly, make notes, select the techniques you like most and train yourself calmly and always taking your time. This will help your mind to learn everything with much more ease and less effort.

Enough said: now I can't do anything but wishing you all the best and, of course, to enjoy your reading!

Yamada Takumi

About the authors:

Yamada Takumi and Danilo Lapegna are two Software Engineers working in London as software developers and freelance writers. Their series of books, "The

101 bibles", has been a great sales success in the self-publishing sector in Italy, engaging thousands of electronic readers even in a moment of economic instability

for the Italian market. Now they're constantly expanding their series with new books about personal and professional self-development and their work represents

an important reference point in the market.

For any kind of questions, help requests or, most of all, feedbacks, you can write them at danilolapegna-101bibles@yahoo.it

We also constantly improve our work with your feedbacks, so don't hesitate to send us anything about possible mistakes, typos or improvements suggestions. As always,

we'll seriously take into consideration the opinions of our readers and we'll use them to enrich our next editions, adding a special mention for your help as well, if

you like so. Turn even on the automatic Kindle Updates for your device to automatically receive our next updates for this ebook!

Thank you for buying "The Speed Math Bible", and thank you for becoming part of the "101 bibles" family!

"The moving power of mathematical invention is not reasoning, but imagination."

II - 5 mathematical strategies that will seriously improve your life

We constantly count and make measurements in our everyday life: how many hours must we sleep to feel really refreshed, what's our ideal body fat percentage, how much time do we need to finish examining those documents, after how many miles the gas tank of our car will be empty ... these are all examples of everyday problems we constantly have to deal with, and that obviously would require a solution as more effective as possible.

At the same time, however, we too often evaluate the factors involved in our problems from a purely qualitative point of view rather than a quantitative one , giving them purely empirical solutions. That is, for example, instead of trying to understand the proper amount of sleeping hours for our body, we too often try to sleep enough to feel relatively fit. Rather than calculating our body fat percentage and adjusting our diet accordingly, we prefer some kind of homemade diet that does not really pay attention to our real physiological needs. And we do that despite it is intuitive that addressing these issues from a quantitative point of view, and so measuring, evaluating, calculating the quantities involved, would offer a greater efficiency to our actions, and so would give us the ability to produce much more with many less expenses in terms of time, money and resources.

Wait: this does not mean that you should run into some weird obsession for rational assessments or for measuring every single aspect of reality. Your existence of course needs even impulsively done actions, lessons learnt from your mistakes and the coexistence with the inevitable unknown.

stronger mathematical "grip" on it could be a great strategy to enhance your personal improvement. After all, the most successful companies are exactly those doing the best in retrieving and analysing data about the consequences of their behaviour. But how could we do that? Here's some advices we could follow for the purpose:

Play giving marks more often. No, this has nothing to do with school. If, for

example, you really enjoy doing something, like, I don't know, eating a particular type of food or going traveling somewhere, try giving to these actions, or to the advantage you get from doing them, a mark. From 0 to 10, from 0 to 100, it doesn't matter! The point is playing making quantitative concepts out of qualitative ones, and starting making better analysis on their basis. For example, if dedicating yourself to jogging in the morning has for you a higher mark than going to gym, you could simply start to decide to ... go jogging more often!

Quantity in most of cases helps you to better focus on the most important things, and cleans your thoughts from the unnecessary stuff!

Try to make measurements where nobody else would do it. "Not everything

that can be counted really counts, and not everything that counts can be really counted", said Albert Einstein.

This phrase not only confirms what I earlier said about the fact that mathematics and calculations sometimes must simply be put aside (luckily), but can help us to understand that, too much often, the most commonly used measurement systems can deceive us, and can divert our attention from the things that really matter. An example I love doing about this is in the school report cards : their obvious purpose should be giving the measurement of the performance of a student, but we often forget that, on the other side, there could be a teacher that simply has an awful teaching method.

And that's where we should start to "make measurements" where nobody else would do it: letting the students evaluate the quality and the goodness of their teacher's work could push everybody into doing better his/her work, and consequently into producing better results for everybody.

Making measurements, evaluating, focusing on the things that other people tend to ignore, gives you that extra oomph that will produce a significant advantage in any kind of situation!

Increase the quantity and improve the quality of your measurement instruments. Imagine: it's a lazy evening and you decide to go watching a movie

you're hesitating between that new action movie with a lot of old celebrities and that new horror Japanese movie about a dead woman who comes back from the grave. Now you could take your smartphone, read the movie reviews from some famous website and simply choose the movie with the highest rate. But of course you would better estimate the quality of those movies after increasing the quantity of your measurement instruments and, for example, reading movie ratings from many different websites instead than from only one of them.

Now there is a concept that can really come in handy: the mathematical mean that, although is quite widely known, we'll explain in a few words, just to make sure that everybody knows what are we talking about.

If you have multiple quantities, multiple measurements, multiple values (ex. 3, 4 and 5 stars rating for a movie on three different websites), you can sum those numbers (ex. 3 + 4 + 5 = 12) and then divide the result by the number of values you considered (three ratings in this case, so 12 / 3 = 4).

This will give you the mean of those quantities, which is universally considered a s a very precious value since, as in the measurements theory, as in the probability one, it's said to be very close to the "real" value of something (if a real value can philosophically or scientifically exist, of course).

But now, let's add something and let's talk about quality of measurement instruments: how many times happened that a multi-award winning movie just left you asleep in your chair?

That simply means that those measurement instruments (the "official" movie reviews we used to estimate the goodness of the product) weren't as reliable as they were supposed to be. And this really happens a lot of times, doesn't it?

But what's the solution? How could we improve the quality of our instruments? Well, we could, for example, browse just among the opinions of the friends of ours who usually have similar taste to ours. After adopting this strategy we'll automatically notice that even fewer "measurements" will be proven to be much more truthful, and so again that sometimes quantity is a poor concept (we should never trust an opinion only because shared by most of people, after all) and quality should have a heavier weight in our everyday reasoning.

If numbers rule over a situation, you must ally with them or you'll definitely be defeated. Sun-Tzu, the most important strategic book of all times, repeatedly

affirms that victory can only be achieved if one harmonizes itself with the natural constants.

So, if we monthly spend more than how much we earn, bankrupt will soon be much more than an abstract concept, and there are no "qualitative factors" that

can change this. If we build a shelving unit whose shelves are structurally made to sustain a heavier weight than the decorative object we just bought, we should simply avoid putting it there.

All the fields strictly ruled by physical, economic, natural rules are clear examples of contexts in which you just can't escape from "numbers supremacy". So, making extra measurements for your shelf, checking your monthly expenses, or being sure that your self-employing project will sustain your rent cost in the long-term are all necessary actions, that not only will definitely improve the quality of your work, but will also help you to avoid disasters.

Yes, this may sound trivial sometimes, but at the same time we should pay attention to the fact that too often we tend to suffer for ignoring exactly the most obvious things.

Doubtful? Use the power of the gain-loss ratio! Suppose that you just received

multiple job offers, with different pro and cons, and suppose to be totally doubtful about which one could be the best choice for your career progression. Or maybe you're looking for a new house, you have various options and you want to understand which one will be more suitable for your needs.

Well, in these cases you could put yourself into the hands of the gain-loss ratio: try to make a list of the cons of each choice, and give to each disadvantage a damage rating between 1 (almost annoying) and 10 (death). And be careful: each choice must have at least a disadvantage with a damage rating greater than 0. For example:

Flat 1:

Very expensive: 7 (pretty big deal)

No smoking allowed: 5 (annoying but almost acceptable)

No pets allowed: 8 (this is going to be a serious problem if I want to buy a dog)

Flat 2:

Very dirty: 9

Very far from the city centre: 8

Now, for each choice, calculate the mean of the damage ratings by its disadvantages, and call the result "damage rating" of that specific choice. In our example it would be:

7 + 5 + 8 = 20 / 3 = Damage rating: 6.66

Flat 2:

9 + 8 = 17 / 2 = Damage rating: 8.5

Now it's time to start considering the bright side of the medal: make a list of the pros for each choice and give to each advantage a "convenience rating". Now, calculate the "convenience rating" of every specific choice by calculating the averages of the convenience ratings of its advantages. In our decision about which apartment going to live in, this could be:

Flat 1:

Very bright: 7

Rooms are pretty clean: 7 Very near to my workplace: 8

Convenience rating: 7 + 7 + 8 = 7.3

Flat 2:

Very cheap: 8 Pets allowed: 8

Convenience rating: 8

You're almost done. Now, for each choice, just divide the convenience rating by the damage rating, and you'll have your gain-loss ratio: an approximate measurement of which choice is (possibly) the best to take:

Flat 1:

Gain-loss ratio: 7.3 / 6.66 = 1.096

Flat 2:

Gain-loss ratio: 8 / 8.5 = 0.94 Flat 1 wins!

Of course this doesn't have to be taken as the ultimate truth, but it can be really useful to show you a brighter way in your decision making process. In addition to this, remember that more honest you will be with yourself in your rating, and more trustworthy will be this index in showing you the best choice to take.

Last important thing: when you're dealing with a situation in which fortuity has a strong influence, this index loses most of its usefulness, and in that case the best thing to do is taking your decision with the help of other factors we'll explain in Chapter XIX.

This chapter ends here, but your journey into the practical and strategic applications of mathematics just started. Most of calculation techniques shown in the next pages, in fact, will often come along with their more useful, and funny applications for everyday life, like probability theory applied to gambling or game theory applied to poker. So, if you continue reading, you will be page after page more surprised and wondering why your teachers only taught you the poorest side of this matter.

III - 4 basic steps to boost your calculation skills

Here I'll start explaining you four helpful fundamentals for improving the speed of your mathematical skills that you'll absolutely need to learn before starting to practise with any creative calculation strategy. Once you learnt them, the 80% of the hard work is done, so ... let's start!

1 - Take the right attitude

Yes, influence of the mental attitude in math is something as underestimated as powerful. If you subconsciously keep repeating yourself that math is not part of your world, that it's too difficult for you and that you will never truly use it in your everyday life, you can be sure that you will never do any progress as well. Rather:

Remember that, whatever is your opinion about your "innate predisposition" to mathematics, you're not obligated to follow it. Yes, definitely there are people that are cut out for mathematics and logic since they were born, and people that had a panic attack in front of their first number written on a blackboard, but our nature doesn't have to be our curse. For example Gert Mittring, who won the gold medal at the MSO mental calculation for nine consecutive years, at school was one of the worst math student in its class. And, before him, many other geniuses like Albert Einstein or Thomas Alva Edison, achieved awful marks exactly in the same subjects today are considered as reference points for. Edison was even called a "mentally retarded boy" by one of his teachers.

So, 1) Our school grades will never be a real measurement of our real skills and, 2) From the adaptive nature of our brain inevitably follows that we, and only we

can decide which part of our mind to cultivate, and which part to leave withering and decaying. And I could bet any amount of money on the fact that, if you properly work with the strategies I'll show you, you will develop skills you thought you never had!

Keep yourself motivated! Motivation is the foundation of every personal growth and self-improvement path! In fact, if we keep ourselves motivated, every barrier becomes nothing but an opportunity to learn, build and improve our talent and creativity. But how do we keep motivated in front of a potentially tedious path made up of numbers and theorems? By learning how to convert every arithmetical problem into a deeply creative challenge! By constantly remembering ourselves that backing out from a mental challenge means gradually losing our brightness and cleverness! By understanding that mathematics can sharpen and empower all those "mind tools" that over and over again will make the difference in our life between profit and loss, happiness and unhappiness, success and failure. And, last but not least, by being aware that an improved ability to "play" with numbers can even impress the people we know, as we'll better see later. This will inevitably help us to build our self-confidence and will improve as our relationship with the others, as our connection with our inner self.

Train yourself. Stop using your smartphone even to perform a "7 x 6" and use your everyday challenges to train your brain with mathematics! For example, try to mentally calculate your restaurant bill or your change after buying your daily newspaper! This advice may sound obvious, but at the same time is probably the most effective of all. After all, we should just remember what Aristotle, the ancient Greek philosopher, said: "We are what we repeatedly do. Excellence, then, is not an act, but an habit!".

Don't be stubborn! Yes, keep yourself motivated and trained, but don't stubbornly insist on mathematical problems that you apparently cannot solve, because sometimes the solutions come just as you think of something else.

In fact, even when we abandon a problem solving procedure, our brain often continues working "in background", expands its perspective and then it suddenly becomes able to give us the best solutions to our problems. Famous, moreover, was the expression "Eureka" that Archimedes, according to legend, shouted after finding a good solution for calculating the volume of solid objects while relaxing during a bath. Legend or not, in any case, when the solution to a mathematical

problem or arithmetic just does not seem to arrive, try to take a break and move on. The act of coming back later on hard problems helps us to approach them from a new, wider perspective and, although it may seem counterintuitive, it's the best way to help our brain to do its job.

2 - Respect your brain

One thing we often forget is that brain and body usually give us back nothing more than what we gave them.

So we should just stop looking at them as bare tools to squeeze the best possible result from, and instead we should always see them as important elements of our self to be fed as much as possible in order to let them give their best fruits in return.

So what? A good brain, and therefore good calculation skills comes from the respect for your body, your health and your own biological rhythms first. In particular:

Make physical activity a habit. Some constant workout, in fact, oxygenates our body and brain, improving our reasoning skills and making even more complex thoughts easier to process. No, you don't have to engage yourself in anything strenuous and even one hour a day walking in order to boost your metabolism will be fine.

Oh, and always keep in mind that Dr. Marilyn Albert, a researcher in the field of brain function, has found that adults are more likely to keep a good brain in old age if they are physically active and keep a good cardiovascular health. So ... make a workout plan to keep you always young!

Eat properly. There are dozens of theories about which the exact meaning of "eating properly" should be and, in addition to this, the habits and body of each of us and make us better suitable for specific kind of diets, so finding a way through the different voices is always a quite difficult challenge.

Well, without necessarily having to consult a nutritionist, my simple advice here is to never abandon some elementary common sense rules: variety, don't overdo, always have a proper breakfast, have a good daily amount of antioxidants, don't omit or exaggerate with refined sugar and, most of all, don't forget to insert into your diet a good amount of foods containing phosphorous and B-complex vitamins, great substances to let your brain work at its best. Some examples of foods containing these nutritional substances? Cereals, fish, nuts.

Coffee? Yes, but in the right amount. Yeah, coffee, canned drinks, black and green tea, and anything containing caffeine certainly enhances our cognitive skills, boosts our attention and improves the speed of our neurological reactions. However, if taken in high doses, these drinks just end up being counterproductive, since their hyper-stimulating effect starts compromising our ability to reason as best as we can.

So be careful, and be aware of what should be your daily limit of caffeine. Of course, this limit can be difficult to calculate because it changes depending on factors such as our gender and age; however, paying attention to the signals of your body can be really helpful, so just try to notice after how many cups of coffee, tea or soft drinks you start to feel confused, distracted and nervous.

Relax. Stress causes our adrenal glands to produce excessive amounts of cortisol, a neurotoxic substance that can damage our synapses (the connections among the masses of neurons in our brain). So if you really want to improve your brain skills you always have to leave yourself some time to work on your concerns, to enjoy the things you really like and to eliminate (or at least learn to handle) as more sources of anxiety and stress as possible.

Sleep. We can anywhere read advices about sleeping 8, 9 or 10 hours every night, but the truth is that each of us can have different needs, depending on our physiology and state of fatigue or stress. So do nothing but paying attention to your body signals and give a reasonable priority to your sleep time, since it will greatly boost your brain skills, your ability to focus and even your problem solving and stress management abilities.

3 - Understand the importance of your memory

Any calculation process is basically made up of two different phases: the first one is the calculation itself, while the other one is the memorization of the previous results. And the latter probably represents a crazily critical point for most of people.

Imagine yourself trying to solve without any pen or paper a long multiplication between three-or-more-figure numbers: probably the hardest thing for you will be exactly trying to retain all the digits coming out of the partial multiplications.

So, on the one hand you'll see that the "secret" behind many calculation strategies in this book is in the fact that they're built in order to let us optimize the use of our short-term memory, and on the other hand it follows that your memory represents a key-skill to train if you want to improve your mathematical abilities.

So, try to follow these advices:

Keep yourself focused: This may sound obvious, but focusing on the tasks you're dedicating yourself to will seriously improve your ability to memorize everything involved. So, try to isolate yourself from any potential disturbance source, always force yourself to completely focus on the "here and now" and your memory will get an awesome boost!

Try to memorize only the essential things. If for example you are mentally performing a long addition, don't repeat yourself any digit of the new addends you're getting, but just memorize the partial results you obtain. Same thing with subtraction, multiplication or division: memorize only the bare minimum amount of elements necessary to go ahead in your calculation. You'll realize by yourself that, after adopting this way of thinking, every arithmetic procedure will be much faster and less tiring.

Know yourself. Somebody, for example, is capable to retain something much better into his memory after visualizing it, somebody else after listening to it instead, and so on. In other words, try to understand how, when and where your brain uses to store information in a faster and more efficient way, and then ... always do your best to help it to do its work! Imagine a big, coloured picture of the numbers you're working with if you prefer visualization, and repeat yourself their name if you are an "auditive" person: this will help you to remember them much more easily!

Divide. Your brain works less hard while trying to memorize many combinations made up of few elements, than while trying to memorize one, single set made up of a lot of things. So, especially if you try to operate with very large numbers, try to split them in many shorter numbers made up of a few digits: remembering them later will be as easy as the proverbial pie.

Use the mnemonic major system: The mnemonic major system is probably the most famous and most commonly used mnemonic technique, due to its simplicity and power. In fact, thanks to it, you will be able to memorize even a really long number by turning it into a word or a phrase, which of course is definitely much easier to remember than a sequence of digits.

1. Convert each digit in your number into a consonant using a specific table. 2. Mix the consonants you got with the vowels you need (or some "unassigned" consonants), in order to create a key-word or a key-phrase to remember.

3. When you need to have your number again, you will just have to take your key-phrase, remove the vowels, and re-use your table in reverse. Oh, and by the way, this is your table:

But since I think that examples are always the best way to explain something, let's imagine you have to remember the number 3240191. You have M - N - R - S - T - P (or b) - D (or t).

After mixing those consonants with some vowels you could obtain "Men are stupid". And more stupid or strange is the resulting phrase, and easier it will be for you to remember!

On the contrary, let's imagine your keyphrase is "Black horse". You have B L -hard C - R - S: your number is 95640!

Yes, this technique may sound difficult at the beginning, but a little training can give you awesome results. And consider that it's alone worth the cost of the entire book, since it won't give you only the ability to make any calculation without pen or paper, but will come in handy every time you'll need to remember dates or telephone numbers as well!

Now you have in your "mental toolbox" a lot of extraordinary instruments for improving your "numerical" memory. Treasure those which better work for you, and sharpen them to boost your mathematical skills at their maximum power!

4 - Strengthen your basis

Mathematics is like a "Lego" building structure: the simplest concepts can be combined together in order to shape more complex theories, those theories can be combined in turn to give rise to even more complex ideas, and so on.

So it's obvious that, if the starter bricks are tin-pot, the final structures will always be crumbling and unstable. In other words, if your mathematical basis is not well set in your mind, all the thinking that lays on that basis will be uncertain, slow and unreliable. And this is a hardly critical point, because it's quite underestimated by 99% of people. This, for example, reminds me about a lot of engineers-to-be I was studying with, who very often were reporting bad results in calculus and algebra tests, not because they actually were ignorant about calculus or algebra, but because very, very often they were just making banal arithmetical mistakes.

It's also intuitive that strengthening your basis is definitely a key-skill to accelerate your calculation making: if, for example, you train yourself to instantly recall from your memory all the results of the basic sums from 1+1 to 9+9, you won't lose anymore a single second while trying to perform a "7+8" you need to carry out during a seven-figure sum.

So, my advice here is to work on reinforcing the following concepts in your mind, as banal or obvious they may sound:

The basic number properties. The basic number properties will be often used in this book in "creative" ways in order to build a lot of special speed math strategies, so you could find it very useful to brush up on them:

○ Commutative property of addition and multiplication: changing the order of the operands in any addition or multiplication doesn't change the final result. And so, for example: 3 + 4 + 5 = 5 + 4 + 3 = 5 + 3 + 4 = 12.

○ Associative property of addition and multiplication: changing the order you perform your addition or multiplication with, doesn't change the final result. So, for example, (3 + 4) + 5 which, after looking at the brackets, implies doing 3 + 4 first and then adding 5, won't be different from 3 + (4 + 5), which instead implies calculating 4 + 5 first and then adding 3.

○ Distributive property of multiplication over addition.

That is: if I have an "a x (b + c)", the result will be the same as "(a x b) + (a x c)".

○ Invariantive property of division: given an a / b division, if you multiply or divide both a and b for the same quantity, the final result won't change.

Let's make a simple example and let's imagine you have a 30 / 10 to perform: after dividing both numbers by 10 you'll have 3 / 1, which result obviously is the same as 30 / 10 and equals 3. Same thing if you multiplied both numbers by 2: 60 / 20 in fact still equals 3.

The multiplication has a very similar property, but with one very significant difference: after performing any "a x b" multiplication, the result doesn't change if you multiply "a" by a generic quantity and divide "b" by the same quantity, or vice versa.

So, given for example a 16 x 2, you can halve 16, but then to keep the exact result you'll have to double 2. So 16 x 2 = 8 x 4. Same thing with, for example, 100 x 9: it will give the same quantity as a result as 300 x 3, or 900 x 1.

○ Identity property: Adding or subtracting 0 from a number doesn't change it. The same happens after dividing or multiplying it by 1.

○ Zero property: Any number, if multiplied by 0 will still equal 0. And, in the same way, the result of a multiplication can never be 0 if neither of the factors is 0.

It may be also interesting to remember that this rule implies that you will never be able to divide any number by 0. Let's take in fact a non-null number, like 14. Division is the inverse of multiplication, and so trying to divide 14 by 0 means asking yourself: "Which number equals 14 when multiplied by 0?". And the zero property obviously tells us that this question has no answers.

Dividing or multiplying by 10, 100, 1000 or any other power of 10: This is a pretty simple calculation as well, but brushing up on it is still a good practice, even because some fast multiplication strategies we'll see in the next pages are exactly based on the concept of transforming an "ordinary" multiplication into a multiplication by a power of 10. So:

trailing zeroes as the power of 10 has.

However, if "a" has a decimal point, then you have to shift the point on the right by as many positions as the trailing zeroes of the power of 10 are. So:

5 x 1000 = 5000 (just write three trailing zeroes)

1.28 x 1000 = 1280 (shift the point by two positions to the right and you get 128. But 1000 has three trailing zeroes, so you have to write one more zero to the result in order to complete the multiplication)

567.3 x 1000 = 567300 (shift the point by one zero to the right and you get 5673. But 1000 has three zeroes, so you have to write two more trailing zeroes to the result in order to complete the multiplication)

○ To divide a number "a" by a power of 10, just shift its decimal point to the left by as many positions as the power of 10 has. If the number is an integer, just think as if it had a ".0" after its rightmost digit. And if, during your operation, the point passes over the digit on the extreme left, then write a 0 on its left.

So, for example:

345 / 1000 = 0.345 (shift the point to the left by three positions. The point passes over the 3, so put a 0 on its left)

2 / 1000 = 0.002 (shift the point to the left by three positions. You will have .2 at first, which will become a 0.2. Then, by shifting again, it will be .02 = 0.02. And so on.)

14 300 / 1000 = 14.300 = 14.3 700 000 / 1000 = 700.000 = 700

And a very interesting thing is that mastering this technique makes your life easier even when you have to perform any kind of multiplication or division between numbers having one or more trailing zeroes.

In particular, in the multiplication case, you can just remove any trailing zeroes, perform the multiplication, and then multiply the result by a power of 10 having the same amount of zeroes as those that were removed.

So, for example, 5677 x 300 = 5677 x 3, which must then be multiplied by 100 (two zeroes removed).

And, similarly, 350 x 2000 = 35 x 2, which must then be multiplied by 10 000 (four zeroes removed).

must still remove any trailing zeroes, but then you have to multiply the result by a power of 10 having the same amount of zeroes as those removed from the dividend, and divide the result by a power of 10 having the same amount of zeroes as those removed from the divisor.

So, if for example you must calculate 90 / 6, you can remove the 0 from the 90 first and then calculate 9 / 6 = 1.5. You just removed a zero from the dividend, so now you have to multiply the result by 10 = 15.

If you have to calculate 9 / 60 instead, you must remove the 0 from 60 first, then perform again 9 / 6 = 1.5, and at the end divide the result by 10, getting a 0.15 as a result.

Last case: let's imagine you must calculate 900 / 60. In this case you must remove two zeroes from 900 and one zero from 60, so, after calculating again your good old 9 / 6 = 1.5, you'll have to multiply the result by 100 and then divide it by 10. This obviously is the same as multiplying it by 10, and so the final result is 15 once again.

Multiplication and addition tables from 0 to 9: probably everybody knows these tables, but at the same time very few people are able to recall them from memory instantly and without making any mistakes. So try to brush up on them, and examine in depth all the "harder cells". For example, it is proven that majority of people has problems with multiplication tables from 7 to 9, and the same happens with the sums between digits from 6 to 9.

So, this is where your "4 basic steps to improve your calculation skills" end. Just keep the right attitude, respect your brain, train your memory and strengthen your basis ... you'll enhance your skills in ways you could never believe. And this is just the beginning!

IV - The finger calculator

Everybody knows that one can perform little additions simply by using his or her fingers. It's one of the first things that they taught us in the school, after all, isn't it? But did you know that, by using your fingers, you could perform even some very ... simple multiplications?

If, for example, you have a shocking blackout during a complex calculation, and you can't remember a specific multiplication table, you can use two awesome tricks that will make you able to instantly perform a lot of possible single-digit multiplications. But let's start with the first, really straightforward technique, which takes advantage of the fact that if you sum together the digits of any multiple of 9 (<90), you will always get 9 as a total. More specifically, it lets you immediately perform any multiplication from 9x1 to 9x10. And here's how you can use it:

Turn both your palms towards your face, pointing the extremity of your fingers upwards.

Now associate to each finger a number from 1 to 10, corresponding to the order they have. So, your left thumb will be 1, the left index will be 2, and so on, till the right thumb, which will be a 10.

To perform 9 multiplied by a, simply lower the "a" finger.

The first digit of your result will be equal to the number of fingers before the lowered one. The second digit of your result will be equal to the number of fingers after the lowered one instead.

So, if for example you want to calculate 9 x 2, the number 2 is associated to your left index. If you lower it, you will see that there is 1 finger before the index and 8 fingers after it: 18!

The same if you don't remember the result of 9 x 6: six is associated to your right pinkie, before the right pinkie there are 5 fingers, and so the first digit is 5. After it there are 4 fingers instead and in fact 9 x 6 = 54.

But let's immediately look at the second technique too, useful for retrieving the result of any multiplication from 6x6 to 10x10. Here it is:

Turn again both your palms towards your face, but now point the extremity of your fingers laterally inwards, in order to let the two middle fingers touch each other.

For both hands, associate 6 to the pinkie, 7 to the annular, 8 to the middle finger, 9 to the index, and 10 to the thumb.

If you want to multiply two numbers, let the extremities of the corresponding fingers touch each other, making sure you keep the palms orientation I just described.

The fingers under the point of contact, including the ones in the point of contact itself are the "under" fingers.

The fingers above the point of contact will be the "above" fingers. Multiply the number of the "under" fingers by ten. Call this result "a".

Multiply "above fingers on the left" by "above fingers on the right". Call this result "b".

Sum a and b together.

But since this technique may seem a little bit more complex than the previous one (but still very easy to manage, anyway), let's introduce a little example and let's try to use it to calculate 6 x 7. We will act like this:

Let our left pinkie touch our right annular (or vice versa).

We will have 3 "under" fingers (the two touching fingers plus the pinkie). 3 x 10 = 30.

"Above right" equals 3, while "Above left" equals 4. 3 x 4 = 12. 30 + 12 = 6 x 7 = 42

Second example, 8 x 7:

Our right annular will touch our left middle finger (or vice versa). "Under" equals 5. Multiplied by 10 equals 50.

"Above right" equals 2. "Above left" equals 3 instead. 2 x 3 = 6. 50 + 6 = 8 x 7 = 56

Here comes the fact that this technique can't be used properly if you cannot remember the lower multiplication tables. But this probably it won't be a big problem, considering that most of people have problems right with multiplications by the digits very close to 10, like 7, 8 or 9.

Through your fingers you can even use two little "mathematical" lifehacks that can come in really handy in everyday life:

Measure the length of any object. It's quite simple: measure one of your fingers with a rule, always keep in mind the result and the next time you need to measure any object, and you don't have the proper tools with you, you don't have to do anything but counting how many times the object is longer than your finger, and then multiplying that result by your finger length.

Measure the height of a tree, a building, or any other big object. This technique was used since ancient times: put a person you know the height of next to the big object you want to measure, place your finger next to your eye at such a distance that that it's exactly as high as that person ... and apply the previous technique! The multiplication here is going to be difficult to perform? Don't worry; the techniques in the next chapters will exactly come to your aid!

Now you just got the full basis to enter into the world of the speed math and so it's time to introduce a very fascinating topic, loved as much by the mathematics fans as by the Asian spirituality one: the Vedic Mathematics.

"Mathematics is a great and vast landscape open to all thinking men that adversely joy, but not very suitable for those who do not like the trouble of thinking."

V - Mathematics and Hindu wisdom

Was the beginning of the 20th century, while an influential researcher of Hindu philosophy discovered into its personal copy of a sacred Hindu book, the Atharvaveda, a summarise of very original mathematical rules. More specifically, in that book he found sixteen Sutras (aphorisms of Hindu wisdom) and some corollaries presenting a completely original and much more creative and flexible approach to various mathematical problems.

Since then this "Vedic approach", which vas completely different from any other approach taught in the Occidental world so far, started spreading all over the world. For example today it's pretty renown among the most important USA schools, where it even represents the most prestigious chapter in their teaching plan.

Probably the popularity of the Vedic Mathematics is exactly due to the fact that it commends everyone's creativity and inner expression , making mathematics easier and more attractive for any kind of student. In fact, many of the mathematical stratagems we'll see in the next chapters will be exactly drawn from the Vedic Sutras, or their corollaries, often combining them in order to get fast, creative and extraordinarily effective calculation strategies.

But let's start immediately and let's explain the meaning and the purpose of the "Nikhilam Sutra", which declaims: "All from 9 and the last from 10".

What? Well, it's not cryptic as it sounds, actually: this Sutra reveals nothing but a very fast and efficient method to calculate the distance (or difference) between any number "a" and its higher power of 10 (which is, in the unlucky case you forgot it,

that number made up of a "1" and as many zeroes as the digits of "a". So, for example, will be 10 for 4, 100 for 55, 1000 for 768, and so on).

So it will basically help you to rapidly perform calculations like "1000 - 658", "10 000 - 4530", "1 000 000 - 564 324", and so on. But its power, as we will soon see, doesn't end here.

In any case, to apply the Sutra in its essential form, you can do like this:

Make sure that the subtrahend (the number after the minus) has as many digits as the zeroes of the power of 10 you want to subtract from. For example, you can use it to calculate 100 - 67, 1000 - 345, or 10 000 - 4589.

Starting from left, subtract each digit of the subtrahend from 9 (so, be careful, calculate 9 minus the subtrahend and not the other way round), except the units digit, which must be subtracted from 10. Then, put all the results of these subtractions in order into the final result.

Every time you get "10" as a result (like, for example, it happens when the units digit is 0), set 0 as current result digit and add 1 to the result digit on its left. That's the result of your subtraction!

So, for example, let's try to calculate 10 000 - 4350. We will have that: After subtracting 4 from 9 the first digit in the result (on the left) will be 5 After subtracting 3 from 9, the second digit in the result will be 6

After subtracting 5 from 9, the third digit in the result will be 4

After subtracting 0 from 10, the fourth digit in the result will be 10. Put 0 in the result and carry 1 to the third digit that will then become a 5. Final result: 5650

With the classic method we learnt at school a subtraction like this would have required a much longer and more complex procedure. But here you can deal with the problem just through a few, simple steps, limiting so even the probability to run into an error.

But let's extend the Sutra now, and let's use it to perform different kinds of subtractions, too.

For example, you can use it to faster perform subtractions between numbers having the same amount of digits, and whose minuend (the number before the minus) is positive and made only up of zeroes except for the leading digit. So, for example, you can

apply it to subtractions like "500 - 388", "6000 - 4567", or "70 000 - 43 200". In fact in this case you can:

Apply the Sutra "All from 9 and the last from 10" in order to obtain the difference between the number and its greater power of 10.

Subtract from the result of the Sutra the difference between that power of 10 and your minuend. So, if for example your minuend was 3000, then you will have to subtract 7000 from the Sutra to obtain your final result. If your minuend was 500, then you will have to subtract 500, and so on.

But another example of extension of the possible applications of the Sutra could be in using it to find the difference between a number and the power of 10 having one more zero than its amount of digits. So, for example you can use it to calculate 1000 - 28, 10 000 - 345 or 100 000 - 6 432. In that case, in fact, you will just have to:

Apply the Sutra to the subtrahend.

Add to the result a number made up of a 9 and as many trailing zeroes as the amount of digits of the subtrahend itself (and so, for example, 900 if the subtrahend has two digits, 9000 for three digits, and so on).

It's pretty simple, isn't it? So, if for example you want to calculate your 1000 - 28 you will have to apply the Sutra to 28, obtaining 72, and then banally add 900: 972!

At this point you may have noticed that those last two examples of Sutra extensions came basically from the same line of reasoning, which is, given any kind of a - b, you can:

Apply the Sutra to b.

Find out what's the higher power of 10 for b.

If "a" exceeds that power by a specific amount, add that amount to the final Sutra result.

If "a" is lesser than that power by a specific quantity instead, subtract that quantity from the Sutra result.

20 004 - 3498? Apply the Sutra to 3498 and add 10 004!

9998 - 4432? Apply the Sutra to 4432 and then subtract 2 from the result.

The first practical application of these strategies could consist in using them to rapidly calculate expenses, starting from an initial budget, which in fact is very often a "rounded" number. For example, if before starting a project you had 7000 dollars on your bank account and now, after a month, only 1288 dollars remain, you can apply the Sutra and then subtract 3000 from its result in order to rapidly calculate how much did you spend. So, in this case, the expenses were 8712 - 3000 = 5712 dollars.

But this was just an introduction to the topic and the applications of the Vedic mathematics of course don't end here. In fact, in the next pages, it will come in handy when we'll start talking about fast multiplication.

VI - Assemble, decompose and Blackjack

This chapter will teach you three basic techniques for more rapidly performing additions and subtractions. Then, at the end of the explanations, it will show you a definitely interesting practical application for all of them.

In addition, I'm pretty sure you've used some of these strategies in past yet, even if in an "intuitive" and "automatic" way. But, anyway, making you fully aware of them will help you to exactly understand when to use them, and so to unleash their full potential. So, let's start explaining the first technique, which is very simple and useful, especially if you have to add (or subtract) among many small numbers. It banally consists in pre-emptively summing up the addends, or the subtrahends, in order to get multiples of 10.

For example, let's suppose that you have to calculate a 3 + 4 + 5 + 6 + 7 + 2 + 5 + 1 + 3 + 4 + 3

The first thing you can do is combining the first "three" and the fifth "seven", getting 10 + 4 + 5 + 6 + 5 + 1 + 2 + 3 + 4 + 3.

Now we take the penultimate "four", the last "three", then the third last "three". After being summed together they equal ten, so we'll have 20 + 4 + 5 + 6 + 2 + 5 + 1.

Again, after adding "four", "five" and "one" I get a third 10, and then 30 + 6 + 2 + 5 = 43.

essentially "creative", especially for children. The "Quest for the ten", in fact, can also be fun and easily help the children to have fun with this specific strategy.

But let's rapidly go to the second technique, which is very similar to the previous one, can be used with additions or subtractions of any kind, and consists in "borrowing" an amount from one of the addends (or subtrahends) and adding it to another one of them in order to transform the latter into a multiple of 10.

Let's assume, for example, we want to calculate a 368 + 214. One thing that you can immediately do here is to simplify the calculations by borrowing a "2" from 214 and carrying it to the 368. So you will get a "370 + 212" that can be calculated in two seconds.

Same thing if we have a 967 - 255. We "borrow" a 3 from the 255, getting a much easier 970 - 258. As you may have noticed, since this is a subtraction, here the rules change a little bit: in fact, any quantity borrowed from the subtrahend must then be added to the subtrahend itself, while any quantity borrowed from the minuend must then be subtracted from the subtrahend instead. So ... be careful in this case!

Off topic: in order to perform 970 - 258 you could use a Sutra as introduced in the previous chapter, and this proves that the real point of strength of the "tools" contained into this book is the fact that they can be freely recombined in order to create appropriate calculation patterns for any kind of situation.

The "borrowing rule" also lets us build these easy simplification rules. They of course may seem obvious to somebody, but it's always a good practice to refresh in your mind even what's obvious, so that you will be able to even more quickly recall it from your memory anytime it's needed (exactly as we said in Chapter II).

If you want to add 9, just add 10 and then subtract 1 If you want to add 8, just add 10 and then subtract 2 If you want to add 7, just add 10 and then remove 3 If you want to subtract 7, just subtract 10 and then add 3 If you want to subtract 8, just subtract 10 and then add 2 If you want to subtract 9, just subtract 10 and then add 1

And of course the same reasoning can be applied to numbers like 19, 90, 80, 900, 800, and so on.

So, let's introduce the last technique, that's very simple as well, and will come in very handy anytime dealing with three-or-more-figure numbers: just decompose one of the operands into a sum among its units plus its tens, plus its hundreds, and so on ...

and then operate separately with these parts!

So, when for example you have to calculate a 3456 - 1234, will be much easier for you if you decompose 1234 into 1000 + 200 + 30 + 4 and then calculate, in sequence: 3456 - 1000 =

2456 - 200 = 2256 - 30 = 2226 - 4 = 2222

The human mind in fact is much more rapidly and easily able to perform a longer set of simple calculations than a single and more complex one. And, as we will see in future, this is the basic principle a lot of speed math strategies are based on.

Now that we explained the three fundamental speed math strategies involving addition and subtraction, let's introduce a commonly known and quite interesting practical application for them: the Blackjack "card counting".

If you don't know what I'm talking about, let's start with a little premise: every gambling game is built in such a way that, in the long term, we will always lose our money while the Casino will always take everything dropping out of our pockets. This means that every gambling game has a negative "expected value", as we will explain more in detail in Chapter XIX.

The only exception to this truth is in the BlakJack and in the fact that, if approached with the right strategy ("counting cards", for example), it can have a positive expected value and so it can lead us to a progressive enrichment in the long term.

Furthermore, Blackjack is a very simple game: you, the dealer and the other players are given an initial two-cards hand. It's each player against the dealer. Face cards are counted as ten points. Ace is counted as one or eleven points depending on the player's choice and every other card keeps its "standard" value.

So, in turn, each player can choose whether to "stay" with his/her hand or to have another card, trying to get as closer as possible to 21 points, but without exceeding this value. In fact, who exceeds 21 points automatically loses his/her bet, which goes to the dealer.

So, after each player still in game defined his/her score by deciding to "stay" with his/her hand, it's the dealer's turn. At this point he/she will have to make the same as the other players, deciding card after card if continuing to draw or keeping the hand.

But the only difference here is that he/she's forced to continue drawing until he doesn't make a score of 17 at least. And if he/she will exceed 21, he/she'll have to pay to every player his/her bet. Otherwise he/she will simply have to pay the bet to any player with a higher score than his/hers, and take it from those having a lesser score. Now that I'm sure you know what I'm talking about, let's introduce some good news about card counting:

Counting cards is simple, or better, we'll talk about a very simple card counting method: the so-called "High/Low".

Counting cards is perfectly legal.

If you have been traumatized by some Hollywood movie in which the people who have been discovered counting cards were hardly beaten up, you can sit and relax because it's just Hollywood fiction!

And now, the bad news:

Even if anybody will unlikely touch you when he/she finds out that you're counting cards, it's seriously possible that he/she could kindly invite you to leave the Casino.

You will never be able to count cards when playing at some virtual Casinos: the virtual decks are virtually and repeatedly shuffled, with no chance to build any winning strategy.

In the "physical" Casinos they actually do everything in their power to reduce your positive expected value to the minimum.

So, the question here could be: why should I ever still be interested in this kind of activity? Well, I can think about two answers:

Because training your calculation skills by gambling is infinitely funnier than doing it by reading books.

Because if you fly low and don't exceed in using this technique, in some little Casinos it's still possible to earn some little cash without any risks or too much effort.

Let's start, then! Let's imagine you're sitting at the Blackjack table with a mojito (hoping it's your first one or your mathematical skills definitely won't be at their best). We will start our count from 0 and, as long as we see the dealer drawing the cards from the deck, we will do like this:

When you see a 10, Jack, Queen, King or an Ace, then subtract 1 from your count

When you see 7, 8 or 9, then leave your count as it is

As you can immediately notice, sure it's a very simple count, but it must necessarily be done very fast, and in order to achieve this you could use the first two strategies introduced in this chapter.

So, as far as we keep counting, we must consider that a high total (+6/+9) reveals that the deck is freight with high cards and figures and, as a consequence, it means that we have a strategic advantage against the dealer (who in fact will more easily exceed 21). A low total instead reveals that we have a higher probability to lose against him/her. More in detail, this is an example of a both prudent and effective strategy. Decide how much a reasonably "basic" bet is and:

Count is negative or equal to 0: don't bet Count is +1: bet your "basic" bet

Count is +2 or +3: bet the double of your basic bet Count is +4 or +5: bet the triple of your basic bet Count is +6 or +7: bet the quadruple of your basic bet

Count is +8, +9 or higher: bet the quintuple of your basic bet

So, here ends our digression about this eternally fascinating Casino game. Many more things of course could be said, but unfortunately this is not the right place. Anyway, never forget that you're lucky enough to be born in the digital information age: just a simple Google search and you'll be able to find a lot of interesting tricks to maximize the probability to beat the dealer while playing Blackjack. Good luck!

VII - The magic column

Let's repeat something very important we said in Chapter II: a calculation strategy becomes as faster and more efficient as it's designed to let the brain store as less information as possible.

Let's suppose, for example, we must perform this addition: 989+

724+ 102+ 670+ 112=

If we had to calculate it through the classic column method we traditionally learnt at school, we should separately sum the units first, then the tens and the hundreds at last, trying to bear in mind every partial amount carried to get the final result. And trying to do that without any pen or paper would be totally frustrating, when not impossible. So, as in mathematics as in life, when something doesn't work properly we have to do nothing but changing our strategy, and here is where the "Magic column strategy" comes in our help. But how can we apply it? Let's see it:

Instead of starting from the units, start from the first digit on the left and sum all the numbers on the left column together, making sure you mentally repeat, while you calculate, only the results of the partial sums.

So for example, if you look at the above-mentioned addition, you should mentally perform 9 + 7, then the result + 1, then + 6, etc., but only saying in your mind "9,

16, 17, 23, 24" (and here to faster perform all the partial sums you can easily use the techniques introduced in the previous chapter).

Once you finished performing the sums on the left column, your partial result is 24. Now move to the next column on the right, take its first digit, put it next to your partial result (so now you will have 24_8) and, without touching the result of the previous column, add to the second number of your partial result the other digits of your new column, exactly like you did in the previous step. So, repeating the same procedure as before, you will have: "24_8, (after adding 2) 24_10, (after adding 0) 24_10, (after adding 7) 24_17, 24_18". And only if, like in this case, after performing your sums you get a number that's greater than 10, put in the new partial result only the units and sum the tens to the number on the left. So, in this case your partial result is 24_18 = 25_8.

Move again to the next column on the right and put its first digit next to your previous result (in this case, so, you will have 25_8_9). Then, behave exactly as you did in the previous steps: don't touch the previous result, and sum the digits of the new columns to the number on the right. So you will have: "25_8_9, 25_8_13, 25_8_15, 25_8_17 = 25_9_7 (since we removed the tens digit)". There are no more columns, so you can remove the underscores and 2597 is your final result.

So we definitely got the result of a long sum in an extraordinarily fast and efficient way, keeping in mind only a very small amount of digits and without paying any attention to the partial amounts to carry.

Of course, in case we must put in column numbers made up of a different amount of digits one from each other, then we should:

Align the units with the units, the tens with the tens, the hundreds with the hundreds, and so on.

Using the same technique as the empty spaces (that inevitably will be on the left side of the smaller numbers) were instead leading zeroes. We could also write those zeroes, if it makes the things easier for us.

But let's have another example and let's say we have to calculate 1341 + 450 + 2451 + 888 + 9872. As a first thing, we must align the numbers in a column and, to make the things easier, we can write a leading zero next to each three-figure number:

0450+ 2451+ 0888+ 9872=

Starting from the left we have: "1, 3, 12". So our partial result is 12.

After going to the right we have: "12_3, 12_7, 12_11, 12_19, 12_27". So our partial result is now 147.

After going to the right again we have: "147_4, 147_9, 147_14, 147_22, 147_29". So the other partial result is 1499.

After working on the last column we have: "1499_1, 1499_2, 1499_10, 1499_12". After carrying the 1 twice, the final result will so be 15 002.

Now, show this new ability to your friends, of course without explaining the strategy behind! I'm quite sure that, if you train yourself to perform any partial sum rapidly and without making any errors, the "scenic effect" will be absolutely awesome!

Do you want to pass to the "next level"? Train yourself to: Work even without aligning the numbers

Work on two or more columns at the same time in order to perform even faster calculations.

But let's explain the second point. Let's imagine we have:

561+ 343+ 912+ 134+ 451=

We know how to operate column by column. But if we want to make the things even faster, we could:

Start from the left as before, but now, as a first step, sum directly the two-figure numbers taken from the first two columns. And so beginning with "56, (+ 34 =)

90, (+ 91 =) 181, 194, 239"

Behave as before on the last column, calculating so: "239_1, 239_4, 238_6, 239_10, 239_11". The final result, in fact, is 2401.

This of course is just the next step, and you'll be confident with it only after properly training yourself to rapidly perform any two-figure number sums (even thanks to the strategies we introduced in the last chapter) and the "single-column" version of this technique as well.

"Mathematics is like checkers, in being suitable for the young, not too difficult, amusing, and without peril to the state."